Embedded newform 23.15.d.a.5.14

• Label: 23.15.d.a.5.14
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.14
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(22.9608 + 6.74191i) q^{2} +(-1346.51 + 1553.96i) q^{3} +(-13301.4 - 8548.26i) q^{4} +(81926.2 + 11779.2i) q^{5} +(-41393.6 + 26602.1i) q^{6} +(-122133. + 55776.2i) q^{7} +(-504531. - 582260. i) q^{8} +(78999.3 + 549452. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	22.9608	+	6.74191i	0.179382	+	0.0526712i	0.370190	−	0.928956i	\(-0.379293\pi\)
							−0.190808	+	0.981627i	\(0.561111\pi\)
\(3\)	−1346.51	+	1553.96i	−0.615688	+	0.710542i	−0.974882	−	0.222721i	\(-0.928506\pi\)
							0.359194	+	0.933263i	\(0.383052\pi\)
\(5\)	81926.2	+	11779.2i	1.04866	+	0.150774i	0.645035	−	0.764153i	\(-0.276843\pi\)
							0.403620	+	0.914927i	\(0.367752\pi\)
\(7\)	−122133.	+	55776.2i	−0.148302	+	0.0677271i	−0.488183	−	0.872741i	\(-0.662340\pi\)
							0.339882	+	0.940468i	\(0.389613\pi\)
\(11\)	6.46388e6	+	2.20140e7i	0.331699	+	1.12966i	0.941473	+	0.337088i	\(0.109442\pi\)
							−0.609774	+	0.792576i	\(0.708740\pi\)
\(13\)	−2.08509e7	+	4.56571e7i	−0.332293	+	0.727620i	−0.999857	−	0.0169326i	\(-0.994610\pi\)
							0.667564	+	0.744553i	\(0.267337\pi\)
\(17\)	−4.17838e8	−	6.50169e8i	−1.01828	−	1.58447i	−0.792107	−	0.610382i	\(-0.791016\pi\)
							−0.226170	−	0.974088i	\(-0.572621\pi\)
\(19\)	1.13245e8	−	1.76212e8i	0.126690	−	0.197133i	−0.772114	−	0.635484i	\(-0.780801\pi\)
							0.898804	+	0.438350i	\(0.144437\pi\)
\(23\)	2.63613e9	−	2.15491e9i	0.774234	−	0.632899i
							\(29\)	−6.51351e9	+	4.18598e9i	−0.377598	+	0.242667i	−0.715649	−	0.698461i	\(-0.753869\pi\)
0.338051	+	0.941128i	\(0.390232\pi\)
\(31\)	−2.47111e10	−	2.85181e10i	−0.898172	−	1.03655i	−0.999132	−	0.0416554i	\(-0.986737\pi\)
							0.100960	−	0.994890i	\(-0.467809\pi\)
\(37\)	7.88824e10	−	1.13416e10i	0.830937	−	0.119471i	0.286297	−	0.958141i	\(-0.407575\pi\)
							0.544640	+	0.838670i	\(0.316666\pi\)
\(41\)	6.23067e9	−	4.33353e10i	0.0319925	−	0.222513i	−0.967553	−	0.252669i	\(-0.918692\pi\)
							0.999545	+	0.0301567i	\(0.00960063\pi\)
\(43\)	−3.72448e11	−	3.22728e11i	−1.37021	−	1.18729i	−0.961558	−	0.274603i	\(-0.911454\pi\)
							−0.408651	−	0.912691i	\(-0.634001\pi\)
\(47\)	−9.02154e11			−1.78072			−0.890361	−	0.455256i	\(-0.849548\pi\)
							−0.890361	+	0.455256i	\(0.849548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.14	✓	270
23.14	odd	22	inner	23.15.d.a.14.14	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.14	✓	270	1.1	even	1	trivial
23.15.d.a.14.14	yes	270	23.14	odd	22	inner